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Thread: x^p=e^x

  1. #1
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    x^p=e^x

    Hi, how do I find the intersection point for this given that p is between and not including 2 and 3 ? Thanks !

    wish the textbook had solutions...
    Last edited by DiscreteMathHelp; Apr 7th 2017 at 07:14 AM.
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  2. #2
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    Re: x^p=e^x

    $x^p = e^x$

    $e^{\ln(x) p} = e^x$

    $\ln(x) p = x$

    this has to either be solved numerically or the solution can be put in terms of the Lambert W function.

    $x = -p W\left(-\dfrac 1 p\right)$
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  3. #3
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    Re: x^p=e^x

    Quote Originally Posted by romsek View Post
    $x^p = e^x$

    $e^{\ln(x) p} = e^x$

    $\ln(x) p = x$

    this has to either be solved numerically or the solution can be put in terms of the Lambert W function.

    $x = -p W\left(-\dfrac 1 p\right)$
    Hey, thanks! No.. sorry, I did get to that part, unsure how to solve now to ensure that p is between 2 and 3, so i'd appreciate if you could give me the steps so i can understand further from what yu did?
    thank you so much ! you will be responsible for passing my exam ! I just dont understand the last line, although igto to the second last line ? so please explain ? also, how do you proceed to find p and x when x>0 ?
    Last edited by DiscreteMathHelp; Apr 7th 2017 at 01:43 PM.
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  4. #4
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    Re: x^p=e^x

    Quote Originally Posted by DiscreteMathHelp View Post
    Hey, thanks! No.. sorry, I did get to that part, unsure how to solve now to ensure that p is between 2 and 3, so i'd appreciate if you could give me the steps so i can understand further from what yu did?
    thank you so much ! you will be responsible for passing my exam ! I just dont understand the last line, although igto to the second last line ? so please explain ? also, how do you proceed to find p and x when x>0 ?
    Looking at the plot of this as you vary $p$ is instructive.

    First off there are no real solutions for $x>0$ if $p < e$

    If $p=e$ there is one real solution, $x=e$

    If $p>e$ there are two real solutions

    What class is this for? I'd be very surprised if you are expected to know what the Lambert W function is.

    Most likely you are expected to solve this numerically, or graphically, or something.

    Can you ask your professor?
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  5. #5
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    Re: x^p=e^x

    Oh, thanks, i'll ask when i get back then... just have a lot to revise for and on holiday
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